\[Y_{nt} \sim N(X_{nt}\beta, \tau_{\omega}^2A_{nt}S_{\omega}A_{nt}'+ \sigma_{\epsilon}^2I)\] \(t = 1,...,T\)
\(Y_{nt}\): Observations
\(X_{nt}\): Covariate values
\(n_t\): locations at time \(t\)
\[A_t = C_tS^{-1}_{\omega}\] where \(S_{\omega}\) is \(m \times m\): Gaussian process
\(C_t\) is \(n_t \times m\) with the \(j\)th row and \(k\)th column entry \(exp(−\phi|s_j−s^*_k|)\) for \(j=1,...,n_t\) and \(k=1,...,m\).
\(C_t\) captures the cross-correlation between the observation locations at time \(t\) and the \(m\) locations, \(s^∗_k\) \(k=1,...,m\)
\(\beta \sim N(0,1000)\)
\(\sigma^2_{\epsilon} \sim Gamma(\alpha_\sigma,\beta_\sigma)\)
\(\tau^2_{\omega} \sim Gamma(\alpha_\tau,\beta_\tau)\)
\(\phi \sim Unif(0,T)\)